Precision, imprecision, higher-order probabilities

Higher order probabilities (HOP) deal better with problems faced by precise and imprecise probabilism (PP/ IP).

Weight of evidence

HOPs allow for a principled, information-theoretic account of weight of evidence, which works better than previous proposals within PP and IP.

Reasoning

The approach is computationally feasible and can be implemented with Bayesian Networks.

A rational agent’s (RA) degrees of belief are to be represented by means of a single probability measure defined over every proposition she entertains

Example: fair coin

\[ \mathsf{P}(H) = \mathsf{P}(\neg H)=.5\]

Example: Unknown bias

\[\mathsf{P}(H) = \mathsf{P}(\neg H)=.5 \]

Locality

RA’s credal stance (in a wide, non-technical sense) about a proposition is to be captured by whatever probability (or probabilities) she assigns to it, and does not depend on what probabilities RA assigns to logically independent propositions

Trouble with evidence responsiveness

PP can’t distinguish between (Fair coin) and (Unknown bias) and multiple other cases

Trouble with sweetening

If RA doesn’t know what the bias of the coin is, learning that it now has increased by .001, might still leave RA undecided

Representors

RA’s credal stance towards \(H\) is to be represented by means of a set of those probability measures \(\mathbb{P}\), which are compatible with evidence

(Bradley, 2019; Fraassen, 2006; Gärdenfors & Sahlin, 1982; Joyce, 2005; Kaplan, 1968; Keynes, 1921; Levi, 1974; Sturgeon, 2008; Walley, 1991)

Evidence responsiveness

• Fair coin: as PP

• Unknown bias: all possible probability measures

Indifference and indecision

• Indifference: \(A\) and \(B\) are equally likely

• Indecision: no comparison, no such determination

(Kaplan, 1968)

Independence preservation

• If every member agrees that \(X\) and \(Y\) are probabilistically independent, the aggregated credence should respect this

• fails on linear pooling and PP

• bunch of other limitative results

  (Dietrich & List, 2016)

The IP approach to pooling

Bundle them up into a set of measures!

Stewart & Quintana (2018)

Learning and imprecision

RA’s representor should be updated point-wise:

\[ \mathbb{P}_{t_1} = \{\mathsf{P}_{t_1}\vert \exists\, {\mathsf{P}_{t_0} \!\in \mathbb{P}_{t_0}}\,\, \forall\, {H}\,\, \left[\mathsf{P}_{t_1}(H)=\mathsf{P}_{t_0}(H \vert E)\right] \}. \]

MaxEnt as our guide

Start learning with a distribution that—given the evidence available—is maximally noncommittal with regard to missing information

(Jaynes, 2003; Williamson, 2010)

Supervaluationist comparison

RA is more confident of \(A\) than \(B\) just in case all members of RA’s representor prefer \(A\)

Two biases

\(\mathsf{P}_1, \mathsf{P}_2\) such that \(\mathsf{P}_1(H)=.4\) and \(\mathsf{P}_2(H)=.6\)?

Two unbalanced biases?

• contains only green marbles

• no information about

Intuitions here

• RA should be certain that the marble drawn from will be green (\(G\)),

• RA should be more confident about \(G\) than that the marble from will be green (\(M\))

The trouble for IP

• For each \(r\in [0,1]\) RA’s representor contains a \(\mathsf{P}\) with \(\pr{M}=r\)

• Including the one with \(\pr{M}=1\)

• So it is not the case that for any of RA’s representor \(\mathsf{P}\), \(\mathsf{P}(G) > \mathsf{P}(M)\)

• So—on IP—RA does not prefer \(G\) over \(M\)

  (Rinard, 2013)

Setup and intuition

Trouble for PP

By (PIE), \(\mathsf{P}_0(H)=.5\) in Stage 0 updates to \(\mathsf{P}_1(H)=.5\) in Stage 1

Trouble for IP

(Levi, 1980)

Trouble for IP

(Levi, 1980)

Trouble for IP

(Levi, 1980)

Precursors

• Beans from a bag, two colors, same observed proportion, different sample sizes (C. S. Peirce, 1872).

• The notion of weight: balance might remain the same while the amount of relevant evidence shifts (Keynes 1921).

Desiderata

• Balance undetermination Different weight with the same balance are possible.

• Weak (strong) increase In Bernoulli trials, weight does not decrease (increases) with sample size keeping frequency fixed.

• Frequency monotonicity In Bernoulli trials, keeping sample size fixed, weight does not decrease as observed frequency goes further from .5.

Weatherson, Joyce, Runde

• Straight flush has the probability of \(\frac{40}{2,958,960}\).

• The player starts behaving confusingly and bluffing.

Hamer’s certainty

Hamer’s absolute distance from 1 or 0 depending on the balance fails at Balance undetermination.

Good’s desiderata and weight

• \(W(H:E)\) is some function of \(\mathsf{P}(E\vert H), \mathsf{P}(E\vert \neg H)\)

• \(\mathsf{P}(H \vert E) = g[W(H:e), \mathsf{P}(H)]\)

• \(W(H: E_1 \wedge E_2) = W(H:E_1) + W(H:E_2 \vert E_1)\)

\[W(H:E) = \log \frac{\mathsf{P}(E \vert H)}{\mathsf{P}(E\vert \neg H)}\]

Good’s own example (expanded)

• a die is selected at random from nine fair dice and one with bias \(\frac{1}{3}\).

• Uniform prior gives you weight of evidence for the loaded die \(log_{10}(.1)\), that is -1 (-10 db).

• Every time you toss it and obtain a six, you gain \(log_{10}(\frac{\frac{1}{3}}{\frac{1}{6}})= log_{10}(2)\)

• Every time you toss it and obtain something else, the weight changes by \(log_{10}(\frac{\frac{2}{3}}{\frac{5}{6}})= log_{10}(.8)\).

Kyburg’s Evidential Probability

\(\mathsf{EP}(H \vert E \wedge K) = [x,y]\)

• Sharpening by richness (prefer frequencies from full joint distributions)

• Sharpening by specificity (prefer proper subsets)

• Sharpening by precision (pick single subinterval if it exists, otherwise, shortest possible cover of minimal subintervals)

Pedden’s weight

Let \(\mathsf{EP}(H \vert E \wedge K) = [x,y]\), then:

\(\mathsf{WK(H\vert E\wedge K)} = 1 - (y-x)\).

• Pedden picks edges by error margins (sensitivity!)

• Also, sensitivity to what happens around the edges only.

• How to deploy outside of combinatorial or frequentist contexts?

• Reasoning with intervals hard to model sensibly (does not preserve structural information)

Challenges to precise probabilism

• insufficient responsiveness to evidence

• fails to model indifference as sensitive to sweetening

• can’t distinguish between lack of knowledge and knowledge that \(\mathsf{P}(X)=.5\)

• trouble with aggregation methods (independence preservation etc.)

Representor with pointwise Bayesian learning

\(\mathbb{P}_{t_1} = \{\mathsf{P}_{t_1}\vert \exists\, {\mathsf{P}_{t_0} \!\in \mathbb{P}_{t_0}}\,\, \forall\, {H}\,\, \left[\mathsf{P}_{t_1}(H)=\mathsf{P}_{t_0}(H \vert E)\right] \}\)

Joyce: \(w(X,E) = \sum_x \vert c(ch(X) = x \vert E) \times (x - c(X\vert E))^2 - c(ch(X) = x) \times (x - c(X))^2\vert\)

• Unintuitive behavior around chance hypotheses

• Failure of weak increase

• no real use of representors

• need to use of distributions over chance hypotheses

• taking credence to be the expected value is non-trivial

Still not evidence sensitive

You still need to go higher-order to model some cases (e.g. uneven bias).

Belief inertia

Point-wise updating can’t make you leave the set of all possible measures.

Unclear mechanism of evidential constraints

• “Drop measures excluded by the evidence”. But how (other than degenerate cases)?

• How exactly does non-testimonial evidence of chances \(\{ \mathsf{P}(X) = x\}\) or \(\mathsf{P}(X) \in [x,y]\) is supposed to arise?

Wrong comparative predictions (Rinard)

• contains only green marbles, no information about

• A marble will be drawn at random from each. You should be certain that the marble drawn from will be green (\(G\)), and \(G>M\).

• IP: for each \(r\in [0,1]\) your representor contains a \(\mathsf{P}\) with \(\mathsf{P}(M)=r\).

• But then, it also contains one with \(\mathsf{P}(M)=1\).

• So not for all \(\mathsf{P}\): \(\mathsf{P}(G) > \mathsf{P}(M)\), \(G\neq M\)!

Proper scoring rule is impossible

See results by Seidenfeld 2012, Mayo-Wilson 2016, Schoenfield 2017, Cambell-Moore 2020

Aggregating doesn’t fly far

• Taking unions leads to skepticism. What else?

• Can’t model synergy

Key idea

Often, uncertainty is not a single-dimensional thing to be mapped on a single one-dimensional scale such as a real line. It is the whole shape of the whole distribution over parameter values that should be taken under consideration. Summaries are just that.

• Much more evidence sensitive (and can go higher order if needed).

• Seamless integration with Bayesian statistics explains learning.

• Belief inertia does not arise.

• HPDI comparison avoids Rinard’s objection.

• There is a proper scoring rule (Urbaniak 2022).

• Evidence-based aggregation is accuracy-wise better than averaging and can model synergy.

• \(m=8\) possible destinations can be reached by making decisions at \(\log_2(8)=3\) forks.

• Surprise: \(1/\mathsf{P}(x)\)

• Shannon information: $ log_2(\mathsf{surprise)) = - log_2((x)) $

• Entropy is average Shannon information:

\[H(X) = \sum \mathsf{P}(x_i) \log_2 \frac{1}{\mathsf{P}(x_i)} = - \sum \mathsf{P}(x_i) \log_2 \mathsf{P}(x_i)\]

(the expected amount of information you receive once you learn what the value of \(X\) is).

Say events arise according to a distribution \(\mathsf{P}\) but we predict them using a distribution \(\mathsf{Q}\).

Cross-entropy

\[\mathsf{H}(\mathsf{P}, \mathsf{Q}) = \sum \mathsf{P}_i \log_2(\mathsf{Q}_i)\]

Kullback-Leibler divergence

\[\mathsf{KLD}(\mathsf{P}, \mathsf{Q}) = H(\mathsf{P}, \mathsf{Q}) - H(\mathsf{P})\\ = - \sum \mathsf{P}_i \log_2(\mathsf{Q}_i) - \left( - \sum \mathsf{P}_i \log_2 \mathsf{P}_i\right) \\ = - \sum \mathsf{P}_i\left( \log_2 \mathsf{Q}_i - \log_2\mathsf{P}_i\right)\\ = \sum \mathsf{P}_i\left( \log_2 \mathsf{P}_i - \log_2\mathsf{Q}_i\right)\\ = \sum \mathsf{P}_i \log_2 \left( \frac{\mathsf{P}_i}{\mathsf{Q_i}}\right) \]

Key idea

The more informative a piece of evidence is, as compared to the uniform distribution, the more weight it has, on scale 0 to 1

\[\mathsf{w(P_i)} = 1 - \left( \frac{H(\mathsf{P})}{H(\mathsf{uniform})}\right)\]

The higher-order approach

• Leads to more honesty in uncertainty assessment

• Is more sensible than sensitivity analysis

• Integrates with Bayesian data analysis

• Leads to an information-theoretic account of evidential weight

• Is computationally feasible

• connections with precise vs. imprecise probabilism in formal epistemology

• problems with existing opinion aggregation methods and a higher-order approach

• modeling synergy of multiple sources of information

• further properties of weight

• relation to evidential completeness

Thank you!